Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 18.0

Time: 40.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.86018806536913692 \cdot 10^{113}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;t \le -1.7434504270103252 \cdot 10^{-90}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k}\\ \mathbf{elif}\;t \le 1.7820116289135629 \cdot 10^{-201}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\sin k}{{\ell}^{2}}}}{\sin k}\right)\\ \mathbf{elif}\;t \le 1.12399449523015534 \cdot 10^{203}:\\ \;\;\;\;\frac{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{{\ell}^{2}}{\sin k}\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r103771 = 2.0;
        double r103772 = t;
        double r103773 = 3.0;
        double r103774 = pow(r103772, r103773);
        double r103775 = l;
        double r103776 = r103775 * r103775;
        double r103777 = r103774 / r103776;
        double r103778 = k;
        double r103779 = sin(r103778);
        double r103780 = r103777 * r103779;
        double r103781 = tan(r103778);
        double r103782 = r103780 * r103781;
        double r103783 = 1.0;
        double r103784 = r103778 / r103772;
        double r103785 = pow(r103784, r103771);
        double r103786 = r103783 + r103785;
        double r103787 = r103786 - r103783;
        double r103788 = r103782 * r103787;
        double r103789 = r103771 / r103788;
        return r103789;
}

↓

double f(double t, double l, double k) {
        double r103790 = t;
        double r103791 = -2.860188065369137e+113;
        bool r103792 = r103790 <= r103791;
        double r103793 = 2.0;
        double r103794 = 1.0;
        double r103795 = k;
        double r103796 = pow(r103795, r103793);
        double r103797 = r103794 / r103796;
        double r103798 = 1.0;
        double r103799 = pow(r103797, r103798);
        double r103800 = pow(r103790, r103798);
        double r103801 = r103794 / r103800;
        double r103802 = pow(r103801, r103798);
        double r103803 = cos(r103795);
        double r103804 = l;
        double r103805 = 2.0;
        double r103806 = pow(r103804, r103805);
        double r103807 = r103803 * r103806;
        double r103808 = sin(r103795);
        double r103809 = pow(r103808, r103805);
        double r103810 = r103807 / r103809;
        double r103811 = r103802 * r103810;
        double r103812 = r103799 * r103811;
        double r103813 = r103793 * r103812;
        double r103814 = -1.7434504270103252e-90;
        bool r103815 = r103790 <= r103814;
        double r103816 = r103795 / r103790;
        double r103817 = pow(r103816, r103793);
        double r103818 = r103804 / r103817;
        double r103819 = 3.0;
        double r103820 = pow(r103790, r103819);
        double r103821 = r103820 / r103804;
        double r103822 = r103793 / r103821;
        double r103823 = tan(r103795);
        double r103824 = r103808 * r103823;
        double r103825 = r103822 / r103824;
        double r103826 = r103818 * r103825;
        double r103827 = 1.782011628913563e-201;
        bool r103828 = r103790 <= r103827;
        double r103829 = r103793 / r103805;
        double r103830 = pow(r103795, r103829);
        double r103831 = r103830 * r103800;
        double r103832 = r103830 * r103831;
        double r103833 = r103794 / r103832;
        double r103834 = pow(r103833, r103798);
        double r103835 = r103808 / r103806;
        double r103836 = r103803 / r103835;
        double r103837 = r103836 / r103808;
        double r103838 = r103834 * r103837;
        double r103839 = r103793 * r103838;
        double r103840 = 1.1239944952301553e+203;
        bool r103841 = r103790 <= r103840;
        double r103842 = r103819 / r103805;
        double r103843 = pow(r103790, r103842);
        double r103844 = r103804 / r103843;
        double r103845 = r103844 / r103817;
        double r103846 = r103843 / r103804;
        double r103847 = r103793 / r103846;
        double r103848 = r103847 / r103824;
        double r103849 = r103845 * r103848;
        double r103850 = r103803 / r103808;
        double r103851 = r103806 / r103808;
        double r103852 = r103850 * r103851;
        double r103853 = r103834 * r103852;
        double r103854 = r103793 * r103853;
        double r103855 = r103841 ? r103849 : r103854;
        double r103856 = r103828 ? r103839 : r103855;
        double r103857 = r103815 ? r103826 : r103856;
        double r103858 = r103792 ? r103813 : r103857;
        return r103858;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if t < -2.860188065369137e+113

   1. Initial program 52.3

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Simplified 38.4

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}}\]

   3. Taylor expanded around inf 20.6

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 20.6

      \[\leadsto 2 \cdot \left({\left(\frac{\color{blue}{1 \cdot 1}}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

   6. Applied times-frac 20.6

      \[\leadsto 2 \cdot \left({\color{blue}{\left(\frac{1}{{k}^{2}} \cdot \frac{1}{{t}^{1}}\right)}}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

   7. Applied unpow-prod-down 20.6

      \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

   8. Applied associate-*l* 18.9

      \[\leadsto 2 \cdot \color{blue}{\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)}\]

   if -2.860188065369137e+113 < t < -1.7434504270103252e-90

   1. Initial program 30.4

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Simplified 23.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 23.6

      \[\leadsto \frac{\frac{2}{\frac{{\color{blue}{\left(1 \cdot t\right)}}^{3}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   5. Applied unpow-prod-down 23.6

      \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{1}^{3} \cdot {t}^{3}}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   6. Applied times-frac 20.8

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{1}^{3}}{\ell} \cdot \frac{{t}^{3}}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   7. Applied *-un-lft-identity 20.8

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{\frac{{1}^{3}}{\ell} \cdot \frac{{t}^{3}}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   8. Applied times-frac 20.7

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{1}^{3}}{\ell}} \cdot \frac{2}{\frac{{t}^{3}}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   9. Applied times-frac 14.9

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{{1}^{3}}{\ell}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k}}\]

   10. Simplified 14.9

       \[\leadsto \color{blue}{\frac{\ell}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k}\]

   if -1.7434504270103252e-90 < t < 1.782011628913563e-201

   1. Initial program 62.1

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Simplified 62.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}}\]

   3. Taylor expanded around inf 25.2

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 25.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{\left({k}^{\left(\frac{2}{2}\right)} \cdot {k}^{\left(\frac{2}{2}\right)}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

   6. Applied associate-*l* 18.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied unpow2 18.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\sin k \cdot \sin k}}\right)\]

   9. Applied associate-/r* 18.1

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{\sin k}}{\sin k}}\right)\]
   10. Using strategy rm

   11. Applied associate-/l* 18.3

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\sin k}{{\ell}^{2}}}}}{\sin k}\right)\]

   if 1.782011628913563e-201 < t < 1.1239944952301553e+203

   1. Initial program 42.9

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Simplified 37.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}}\]
   3. Using strategy rm

   4. Applied sqr-pow 37.1

      \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   5. Applied times-frac 24.9

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   6. Applied *-un-lft-identity 24.9

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   7. Applied times-frac 24.9

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\]

   8. Applied times-frac 18.1

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}{\sin k \cdot \tan k}}\]

   9. Simplified 18.1

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}{\sin k \cdot \tan k}\]

   if 1.1239944952301553e+203 < t

   1. Initial program 57.2

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Simplified 42.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}}\]

   3. Taylor expanded around inf 22.1

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 22.1

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{\left({k}^{\left(\frac{2}{2}\right)} \cdot {k}^{\left(\frac{2}{2}\right)}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

   6. Applied associate-*l* 22.1

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 42.6

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\left(\sqrt{\sin k} \cdot \sqrt{\sin k}\right)}}^{2}}\right)\]

   9. Applied unpow-prod-down 42.6

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{{\left(\sqrt{\sin k}\right)}^{2} \cdot {\left(\sqrt{\sin k}\right)}^{2}}}\right)\]

   10. Applied times-frac 41.9

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{\cos k}{{\left(\sqrt{\sin k}\right)}^{2}} \cdot \frac{{\ell}^{2}}{{\left(\sqrt{\sin k}\right)}^{2}}\right)}\right)\]

   11. Simplified 41.9

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\color{blue}{\frac{\cos k}{\sin k}} \cdot \frac{{\ell}^{2}}{{\left(\sqrt{\sin k}\right)}^{2}}\right)\right)\]

   12. Simplified 20.9

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \color{blue}{\frac{{\ell}^{2}}{\sin k}}\right)\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.86018806536913692 \cdot 10^{113}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;t \le -1.7434504270103252 \cdot 10^{-90}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k}\\ \mathbf{elif}\;t \le 1.7820116289135629 \cdot 10^{-201}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\sin k}{{\ell}^{2}}}}{\sin k}\right)\\ \mathbf{elif}\;t \le 1.12399449523015534 \cdot 10^{203}:\\ \;\;\;\;\frac{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{{\ell}^{2}}{\sin k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))